A pipe used to convey fluid is cantilevered from a wall. The steel pipe \(\left(ho=7500 \mathrm{~kg} / \mathrm{m}^{3}, G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, E=200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) has an inner radius of \(20 \mathrm{~cm}\), a thickness of \(1 \mathrm{~cm}\), and a length of \(4.6 \mathrm{~m}\). For an empty pipe determine

(a) The five lowest natural frequencies for torsional oscillation.

(b) The five lowest natural frequencies for longitudinal vibration.

(c) The five lowest natural frequencies for transverse motion.